Unit vectors

I was wondering why we use unit vectors, what's happens if we don't and in the definition of a unit vector, does a magnitude of 1 mean a scalar of 1 (for a that vector) or that the vector itself only represents one/single arrow ?

We can make all of our equations without unit vectors (but not without vectors), just divide and multiply by the magnitude.
In terms of the magnitude of a unit vector (or any vector) it relates to size.

A vector is something that carries two pieces of information, direction and size. So 3km east is a vector, or 1 Newton down. We often use unit vectors when we are supplying the size information elsewhere so we'll have The maginutde (a scalar or number, it carries the units, too) the direction (in english this means 'in the direction that you measured r') It still carries a magnitude (1), but this is replaced by the other magnitude when they are multiplied (1x something=something).

Another unit vector would be 1 unit east, or 'in the direction I drove'
so you could have 5 km (magnitude) 'in the direction I drove'(unit vector), together they make a normal vector.

It means vector's length is 1. If you know the dot product, it means that . If you don't, it means that if you take a ruler and measure the vector, it'll be 1 unit long.

It's convenient because you can do math with unit vectors without changing the length of your original vector. For example, in physics we might do:

where is a unit vector pointing in the same direction as . In this equation, would be a vector pointing from one object of mass to another object of mass , and would be the gravitational force between them.

If we don't use a unit vector in the equation, we end up multiplying by a vector of some random length, and we make the gravitational force vector too strong or too weak.

Those certainly are not the only notations used, even for three space. You will see to denote the unit vectors along the x, y, and z axes, for the spherical unit vectors, and more generically to denote an arbitrary set of unit vectors. That's just a starter.

And of course i, j, and k only pertain to three space.

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This is modern software; there's no useful manual. After all, changing how everything works every six months or so is more important than helping people use the features you have.